In this paper, we examine the logics of essence and accident and attempt to ascertain the extent to which those logics are genuinely formalizing the concepts in which we are interested. We suggest that they are not completely successful as they stand. We diagnose some of the problems and make a suggestion for improvement. We also discuss some issues concerning definability in the formal language.

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics, and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics and in speech act theory. As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation (...) of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion. (shrink)

This article outlines a semantic approach to the logics of unknown truths, and the logic of false beliefs, using neighborhood structures, giving results on soundness, completeness, and expressivity. Relational semantics for the logics of unknown truths are also addressed, specically the conditions under which sound axiomatizations of these logics might be obtained from their normal counterparts, and the relationship between refexive insensitive logics and logics containing the provability operator as the primary modal operator.

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor (...) sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

We present a generalization of the algebra-valued models of \ where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of designated values. Under this generalization there are many algebras which are neither Boolean, nor Heyting, but that still validate \.

In this paper we study a new operator of strong modality ⊞, related to the non-contingency operator ∆. We then provide soundness and completeness theorems for the minimal logic of the ⊞-operator.

We offer tableaux systems for logics of essence and accident and logics of non-contingency, showing their soundness and completeness for Kripke semantics. We also show an interesting parallel between these logics based on the semantic insensitivity of the two non-normal operators by which these logics are expressed.

In this paper, we argue for an instrumental form of existence, inspired by Hilbert’s method of ideal elements. As a case study, we consider the existence of contradictory objects in models of non-classical set theories. Based on this discussion, we argue for a very liberal notion of existence in mathematics.

In this article we analyze the method of forcing from a more philosophical perspective. After a brief presentation of this technique we outline some of its philosophical imports in connection with realism. We shall discuss some philosophical reactions to the invention of forcing, concentrating on Mostowski’s proposal of sharpening the notion of generic set. Then we will provide an overview of the notions of multiverse and the related philosophical debate on the foundations of set theory. In conclusion, we connect this (...) modern debate and Mostowski’s proposal, suggesting a way to analyze the notion of genericity within the framework of a multiverse structure. (shrink)

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

In the Tractatus, it is stated that questions about logical formatting cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of logic, (...) that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict. (shrink)

We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of König, Larson, Moore and Veličković and simplifies the original proof of Moore.

Je me propose dans cet article de traiter de la théorie des ensembles, non seulement comme fondement des mathématiques au sens traditionnel, mais aussi comme fondement de la pratique mathématique. De ce point de vue, je marque une distinction entre un fondement ensembliste standard, d'une nature ontologique, grâce auquel tout objet mathématique peut trouver un succédané ensembliste, et un fondement pratique, qui vise à expliquer les phénomènes mathématiques, en donnant des conditions nécessaires et suffisantes pour prouver les propositions mathématiques. Je (...) présente quelques exemples de cette utilisation des méthodes ensemblistes, dans le contexte des principales théories mathématiques, en termes de preuves d'indépendance et de résultats d'équiconsistance, et je discute quelques résultats récents qui montrent comment il est possible de « compléter » les structures H et H. Ensuite, je montre que les fondements ensemblistes de mathématiques peuvent être utiles aussi pour la philosophie de la pratique mathématique, car certains axiomes de la théorie des ensembles peuvent être considérés comme des explications de phénomènes mathématiques. Dans la dernière partie de mon article, je propose une distinction plus générale entre deux différentes espèces de fondement : pratique et théorique, en tirant quelques exemples de l'histoire des fondements des mathématiques.In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H and H. Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

We review Linnebo's Philosophy of Mathematics, briefly describing the content of the book.

In this paper, we investigate the family LS0.5 of many-valued modal logics LS0.5's. We prove that the modalities of necessity and possibility of the logics LS0.5's capture well-defined bivalent concepts of logical validity and logical consistency. We also show that these modalities can be used as recovery operators.

: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one (...) of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution. (shrink)

This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how (...) Hilbert's conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert's foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert's position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert's for the completeness of set theory. (shrink)

ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the (...) connection between the notions of independence and simplicity. (shrink)